2: Resistors: Ohm's Law, Series and Parallel Combinations

Shaun Bradley Nov 29, 2025

Conductors and Insulators

A conductor is an object that easily conducts electricity, meaning current will easily flow through it when a voltage is applied to it. Likewise, insulators do not easily conduct electricity. For example, a piece of metal will have a very high current flowing through it if even a small voltage is applied between each of its ends, but a piece of wood would not have any current flow though it. However, if enough voltage is applied to the wood, then current will flow though it - just like how lightning can strike a tree and cause current from the lightning bolt to flow through the tree. Enough voltage can overcome any resistance.

Ohm's Law

Ohm's law tells us how much current will flow when a voltage is applied to a material. Each material has its own resistance properties.

In the equation below, V is the voltage (the difference in electrical potential on each side of the resistor), R is the resistance of the resistor (measured in Ohms), and I is the current that flows (measured in Amps) resulting from the applied voltage.

I prefer to write the equation in this form instead of V = IxR because it makes it more clear that it is the voltage that causes the flow of charge (current), but other forms of the equation can be used if they are needed.

Resistors in Series

When connecting resistors, they can be either in series, parallel or neither, with other resistors. When a resistor is in series with another resistor, the same current will flow though both resistors resulting in two separate voltages across each individual resistor.

What we will often want to do is find the equivalent resistance. This means we want to simplify part of a resistor circuit as much as possible into one single resistor. In the circuit above, this equivalent resistance would be the sum of the two resistances. Mathematically speaking, we are asking what single resistance would give the same voltage drop as the two resistors above combined. We can write that the total voltage is the sum of the other two voltages V1 and V2, and also write V1 and V2 using Ohm's law:

We can rearrange Ohm's law so that we are solving for the resistance that would be required for the voltage Vtotal to produce the current I. After this we substitute the equations and solve:

Therefore, when combining resistors in series, they add together.

Resistors in Parallel

A resistor is in parallel with another resistor if it shares the same connection point at both sides. The same currents that left the node going into the resistors are the same currents that join back together after coming out of the resistors.

Resistors in parallel act almost opposite to resistors in series. While resistors in series share the same current but divide the voltage between themselves, resistors in parallel divide the current between themselves but share the same voltage.

This time, we will want to know what single resistance would give the same current draw as the two resistors above. Since V1 and V2 are the same voltage, I will refer to them as V. First I write the total current and then use Ohm's law to find the current in terms of the other parameters.

Again, write the equivalent resistance in terms of Ohm's law and substitute the other equations above.

Therefore, when combining resistors in parallel one cannot simply add them together, but needs to use the equation.

How to know?

How do we know if the resistors are in series or parallel?

1) If one end of a resistor is connected to the beginning of another, so that all the current flows from one resistor to the next, then it is in series.

2) If both ends of the resistor are each connected to one another, so that the current splits between each resistor and the re-joins, then it is in parallel.

For example, just because the resistors are geometrically parallel to each other does not mean that they are electrically parallel:

What happens if a resistor is not in series or parallel with another? In the example below, are resistors R2 and R3 in parallel or series?

The answer is neither. The resistors are only connected together at point B and not on the other side because they go to separate nodes C and G, so they are not in parallel. They are also not in series since all the current from R2 does not go though R3 (or vice versa) because point B has another resistor connected to it.

Manufacturing Tolerance

Resistors will never be the exact value you want them. During any manufacturing process, you can never make exactly the same thing millions of time over due to inacuracy of the machines and variation in the materials used. Because of this, manufacturers will label their resistors with a tolerance. For example, a 1k resistor (1000 Ω) with a 5% tolerance (±5%):

Note: The tolerance is a percentage (per 100).

So, a 1k resistor could be anywhere between 950 Ω and 1050 Ω.

Example 1

Find the value and direction of the current in the circuit. Also label the voltages V1 and V2. Assume R1 and R2 are each 1000 Ω.

First, we know the current is leaving the power source which supplies power. Next, continue to draw the current flowing around the loop (since that is the only way for it to flow). Next, draw the voltage polarities according to the sign convention:

After this, we can use Ohm's law (current equals voltage divided by resistance) to find the current. The two resistors are in series since all the current flowing through R1 flows through R2. We want to use the total voltage over both resistors and the total resistance in order to find the current.

We could have used just V1/R1 or V2/R2, but this would require us to know V1 and or V2; using the sum will be easier. If we do a KVL statement around the loop we have:

Since we now know the value of V1 + V2, we can solve:

Therefore, the current is 2.5 mA (milliamps).

Example 2

Find the voltage at node c. Assume R1 and R2 are 1000 Ω and R3 and R4 are 500Ω.

The procedure will be as follows:

Simplify the circuit to find the current through R1.
Go back to the full circuit and use the current through R1 to find Vb.
Find the current through R3 and R4 by KCL and Ohm's law.
Use the current through R3 and R4 to find Vc.

Step 1:

First, note that R3 and R4 are in series. Create a new series equivalent resistor R5.

Now, we can see that R2 and R5 are in parallel, we simplify this further by making a new parallel equivalent resistor R6.

Now, we use the same method as in example 1 to find the current I1.

Step 2:

Since we want to find Vc, we need to go back to the unsimplified circuit since our simplification eliminated it. It will be helpful to draw the current and voltages on each component.

Now, using KVL from ground to point b we find:

Step 3:

Now we need to find the current through the R3 and R4 branch (labelled I3), then after we can use the same process as step 2 to find Vc. We start with a KCL statement at node b.